The propagator of the finite XXZ spin-$\tfrac{1}{2}$ chain

TL;DR

This paper derives exact contour integral formulas for the finite volume real space propagator of the spin-1/2 XXZ chain, valid for any anisotropy and boundary conditions, using spectral sum and lattice path integral methods.

Contribution

It provides novel exact formulas for the finite volume propagator of the XXZ chain, independent of the Bethe Ansatz solution and string hypothesis, and applies them to quantum quench problems.

Findings

Exact contour integral formulas for the propagator in finite volume.

Application to Loschmidt amplitude for quantum quenches.

Extension to perturbed XXZ models with Dzyaloshinskii-Moriya interaction.

Abstract

We derive contour integral formulas for the real space propagator of the spin-$\tfrac12$ XXZ chain. The exact results are valid in any finite volume with periodic boundary conditions, and for any value of the anisotropy parameter. The integrals are on fixed contours, that are independent of the Bethe Ansatz solution of the model and the string hypothesis. The propagator is obtained by two different methods. First we compute it through the spectral sum of a deformed model, and as a by-product we also compute the propagator of the XXZ chain perturbed by a Dzyaloshinskii-Moriya interaction term. As a second way we also compute the propagator through a lattice path integral, which is evaluated exactly utilizing the so-called $F$-basis in the mirror (or quantum) channel. The final expressions are similar to the Yudson representation of the infinite volume propagator, with the volume entering as a parameter. As an application of the propagator we compute the Loschmidt amplitude for the quantum quench from a domain wall state.